Download Lesson 3

Grade 4 Mathematics
Data Analysis, Probability, and Discrete Mathematics:
Lesson 3
Read aloud to the students the material that is printed in boldface type
inside the boxes. Information in regular type inside the boxes and all
information outside the boxes should not be read to students. Possible
student responses are included in parentheses after the questions.
NOTE: The directions read to students may depend on the available
materials. Read only those parts of the lesson that apply to the materials you
are using.
Any directions that ask you to do something, such as to turn to a page or to
hand out materials to students, will have an arrow symbol () by them.
Purpose of Lesson 3:
• In this lesson, the tutor and the students will
ü understand the meaning of the terms certain, impossible, likely,
unlikely, and equally likely as they apply to probability; and
ü determine the probability of events.
Equipment/Materials Needed:
• Copies of Student Sheets 81 – 83
• Paper and pencils
• Chalkboard
• One coin of any kind and one number cube with the numbers 1, 2, 3, 4, 5,
6 on it
• Paper clips
Preparations before beginning Lesson 3:
• Run one copy of Student Sheets 81 – 83 for each student.
• Have paper and pencils available.
• Have one paper clip per two students available to use on the spinners.
Grade 4 Mathematics
37 A
Lesson 3, Data Analysis
Lesson 3: Data Analysis
 Give Student Sheet 81 to the students.
Say:
It is often useful to know whether something is likely or unlikely to
happen. An event is something that may happen. Sometimes, we are
certain that an event will occur. Sometimes, we know it is impossible;
and sometimes, we just aren’t sure. Let’s look at the sentences in Part
A. I want you to tell me what is the chance that each of these events will
happen. Is it certain, impossible, or maybe? Many of the students will say
that it is certain that they will eat dinner, but something may happen on the
way home and they might miss dinner. They might get sick; dinner might
get burned, etc.
Answers:
1) Maybe
2) Impossible
3) Maybe
4) Certain
5) Maybe
6) Maybe
Let’s look at those events for which you answered “maybe.” Do you
think that these are likely or unlikely to happen? (1. It really depends on
the weather. 3. Probably unlikely for one class, but maybe likely for the
entire school. 5. Likely 6. It is just as likely that we will get a tail. This
event is considered equally likely to happen or not to happen.)
Ask students to tell you some events that are the following:
Certain (The sun will rise.)
Impossible (Frogs will talk.)
Likely to happen (We will have rain next month.)
Unlikely to happen (I will not watch TV for a year.)
Equally likely to happen or not to happen (If I pick a card from a deck of
cards, it will be red.)
Say:
Probability is the likelihood that something may happen. Instead of just
saying something is likely to happen, we can use numbers from 0 to 1 to
describe probability. What kind of numbers have we studied that are
between 0 and 1? (fractions, decimals, and percents) Note: Students may
not say percents. Percents will be introduced in only this lesson.
Grade 4 Mathematics
38 A
Lesson 3, Data Analysis
 Draw the following number line on the board.
Probability Line
|
0
|
¼
|
½
|
¾
|
1
Say:
Probability is always shown as a number from 0 to 1. If the probability
of an event happening is 0, the event is impossible. Tell me an event that
has a probability of 0. (Today is Sunday, so tomorrow will be Thursday.)
How could I write 0 as a decimal? (0.0) Sometimes you will hear the
word percent used when talking about probability. You might say that
the probability of our mathematics books talking is 0 percent. This
symbol is used for percent. (%) We would write 0%. (Write 0% on the
board.) The probability of an impossible event can be represented as 0,
0.0, or 0%
If the probability of an event is 1, then we know the event will happen.
What is an event that has a probability of 1? (A triangle will always have
3 sides.) How could you write this answer as a decimal? (1.0) Sometimes
you will hear the weather announcer say that the probability of rain
today is 100%. That prediction means that the announcer is certain that
it will rain. The probability of an event that is certain to happen can be
represented as 1, 1.0, or 100%.
Often we do not know whether an event is impossible or whether that
event will definitely happen. When we use numbers closer to 0 than to 1,
we are representing the probability of an event that is not likely to
happen. What are some numbers closer to 0 than 1? (Answers will vary:
Possible answers are
1 1
, , 0.1, 0.2, etc.) When we use numbers that are
10 4
closer to 1 than to 0, we are representing the probability of an event that
is likely to happen. What are some numbers closer to 1 than 0? (Answers
will vary: Possible answers are
9 3
, , 0.9, 0.8, etc.) If an event is as likely
10 4
to happen as not, the event has a probability of ½. How can you write ½
as a decimal? (0.50)
Grade 4 Mathematics
39 A
Lesson 3, Data Analysis
 Give the students Part B of Student Sheet 81. On this part, number 8 is
practice for recognizing the decimal representation of 1/4, 1/2, and 3/4. This
representation of decimals will help with understanding percents. Number 9
gives practice on percents. They do not really need to understand percents at
this time. They must know that the smaller the percent, the less likely an
event is to occur.
Answers:
7) 1. This event could be placed almost anywhere on the line except at 0
or 1. The event’s occurrence depends on the weather. But let them talk
about their predictions.
2. This event should placed be at 0; it is impossible.
3. If students think of one class, they may place the event around ¼; if
they think of the entire school, they may put it closer to ¾. Again,
discussion is critical.
4. This event is certain and should be placed at 1.
5. This event is pretty likely, so is could be placed between ¾ and 1.
6. This event is equally likely, so it should be placed at ½.
8) Decimals: 0.25, 0.50, 0.75
Descriptions: unlikely, equally likely, likely
9) Percents: 25%, 50%, 75%
Descriptions: unlikely, equally likely, likely
 Have students talk about Part C of Student Sheet 81.
Answers:
1) Likely
4) Impossible
2) Equally likely
5) Certain
3) Unlikely
Say:
When you toss a coin, say a nickel, two things can happen. These two
things are called outcomes. What are the two possible outcomes? (You
could get a head or a tail.) The probability of tossing a head is ½ and the
probability of tossing a tail is ½. The outcomes are equally likely to
happen.
If you roll a number cube with the digits 1, 2, 3, 4, 5, and 6 on the sides,
how many outcomes would you have? (6) What are the outcomes? (1, 2,
3, 4, 5, or 6) What is the probability of rolling a 5? (There is 1 way to roll
a 5 and there are 6 outcomes, so the probability is
Grade 4 Mathematics
40 A
1
.)
6
Lesson 3, Data Analysis
 Give students Student Sheet 82.
Answers:
1
6
1
5)
4
1)
2)
2
1
or
6
3
6) 0
3) 0
7)
4)
2
1
or
4
2
3
1
or
6
2
8) 1
9)
3
7
Say:
A game is considered fair if each player has an equal chance of winning.
Suppose we were playing a game that used a number cube. If I got a
point every time an even number was rolled and you got a point every
time an odd number was rolled, the game would be considered fair.
Why? (The probability of getting an odd number is
getting an even number is
1
and the probability of
2
1
.) But I could win any time a number greater
2
than one was rolled and you could win only if you rolled a one; the
game, therefore, would not be fair. I would have a better chance of
5
6
winning. What would be the probability of my winning? ( ); Of your
1
6
winning? ( )
 Give Student Sheet 83. Have students play with a partner. They can use a
pencil and a paper clip to make a spinner. (There was not room for
instructions on the sheet, so read the instructions to the students.)
Say:
We are going to play 4 spinner games. One of you will be red, and one of
you will be blue. Each of you should take turns spinning 10 times. Keep
a tally of when blue or red is spun. If you land on a line, spin again. If
you land on another color, “other” gets a tally mark. Is the game fair?
Do the same thing for the other 3 spinners. Spinners 1 and 3 are fair, while 2
and 4 are not.
 Have one student summarize today’s lesson. Understanding probability
helps students make predictions about problems that involve chance.
Grade 4 Mathematics
41 A
Lesson 3, Data Analysis
Student Sheet 81 (Data Analysis: Lesson 3)
Part A
Certain, Impossible, Maybe
1. It will be raining tomorrow.
2. Our mathematics teacher will grow to 10 feet tall this afternoon.
3. Eight students will be absent tomorrow.
4. The day after Sunday will be Monday.
5. We will all eat dinner tonight.
6. If I toss a coin, it will come up heads.
Part B
7. Where would you place the events in Part A on the number line? Mark the
points on the line and label them as 1, 2, 3, 4, 5, or 6.
|
0
|
¼
|
½
|
¾
|
1
8. This number line shows probabilities. Fill in the blanks with the correct
decimal for each marking. Describe the probability of each in words.
|
0
0.0
Decimals:
Descriptions:
|
¼
_____
|
½
_____
|
¾
_____
|
1
1.0
Impossible ________ ________ ________ Certain
9. This number line shows the probabilities but uses percents. Fill in the
blanks with the correct percent for each marking. Describe the
probability of each in words.
Percents:
|
0%
|
|
0%
______
______
|
_______
Descriptions: Impossible _______ _______ _______
Grade 4 Mathematics
42 A
|
100%
100%
Certain
Lesson 3, Data Analysis
Part C
Are the events below impossible, unlikely, equally likely, likely, or
certain to occur?
1. Ms. Compton said the probability of having homework tonight was 75%.
2. Tommy said that the likelihood of his mother’s having a baby girl was
1
.
2
3. Since Tia has been punished this week, she said the chance of her
watching TV tonight was
1
.
4
4. The probability of your mathematics book talking to you tonight is 0.
5. If you pull a card from a standard deck of cards, the probability that you
will pull a red or black card is 100%.
Grade 4 Mathematics
43 A
Lesson 3, Data Analysis
Student Sheet 82 (Data Analysis: Lesson 3)
For problems 1 – 4, pretend you are rolling a number cube with the
numbers 1, 2, 3, 4, 5, and 6 on the faces.
1. What is the probability of rolling a 3? _____________
2. What is the probability of rolling numbers greater than 4? _____________
3. What is the probability of rolling numbers less than 1? ______________
4. What is the probability of rolling an even number? _____________
For problems 5—8, use the spinner below.
Red
Blue
Green
Yellow
5. If you spin the spinner once, what
is the probability of spinning red?
________________
6. What is the probability of spinning
orange? ________________
7. What is the probability of spinning
blue or green? ____________
8. What is the probability of spinning red, blue, green, or yellow? ________
9. Suppose you had cubes in a bag. Three are red, 2 are blue, and 2 are
green. Without looking, you reach in and pull out one cube. What is the
probability that the cube will be red? ____________
Grade 4 Mathematics
44 A
Lesson 3, Data Analysis
Student Sheet 83 (Data Analysis: Lesson 3)
1.
Total
Red
Blue
Red
Blue
Other
2.
Blue
Red
Green
Total
Red
Blue
Other
3.
Total
Blue
Red
Red
Blue
Red
Blue
Other
4.
Total
Blue
Green
Grade 4 Mathematics
Red
Blue
Total
Red
Blue
Other
45 A
Lesson 3, Data Analysis